The Two-Sheeted Topology of Extended Kerr-Type Spacetimes and a Parity-of-Crossings Property for Ring-Traversing Geodesics

We revisit the two-sheeted structure of extended Kerr-type spacetimes. After excising the singular ring, the signed-$r$ oblate point space realises the unique connected meridional double cover of a separate one-sheet ring complement. The cover is unbranched on the excised space; describing the ring as a branch locus refers only to a topological completion and does not regularise the Kerr metric there. Admissible causal geodesics can cross the regular part of the $r=0$ disk without meeting the ring. The Kerr equations give $R(0)=-a^{2}Q$; a transverse crossing requires $Q<0$ together with polar admissibility, and its smooth continuation changes the sign of $r$. Closed projected loops obey intrinsic deck monodromy, whereas open paths obey a cut-relative mod--$2$ label law. Even crossing parity restores the label; odd parity reverses it. For $N$ mutually unlinked rings in a simply connected reference region, the fundamental group is $F_N$. We classify its connected double covers and identify the uniform choice assigning non-trivial monodromy to every meridian. Conditional on a supplied disk-separated continuation, total crossing parity determines the final branch label. In the subextreme maximal Kerr extension, compactness and local finiteness extend the signed-$r$ law to every compact transverse segment, while event- and Cauchy-horizon crossings are neutral. Under separate locally finite reference hypotheses, the Carter chain admits the corresponding uniform cover; no parity at infinity is assigned. Ordinary Kerr disk crossing does not reverse time orientation. If a chosen completion identifies the restricted time-orientation cover with the uniform meridional cover, every smooth closed timelike curve in that reference region has even uniform parity. A conditional Novikov fixed-point law distinguishes ordinary from deck-exchanged return data, but topology does not discretise the classical solution space.